The training of VAE-GAN is a combination of the training procedures of VAE and GAN. We train the encoder and the decoder (generator \(G\) ), with reconstruction loss \(L_R\)  and KL Divergence loss \(L_{KL}\)  from [1]} and the adversarial loss for generator \(L_{adv}^G\) . The adversarial loss for \(G\)  is \(L_{adv}^G = L_{bce}(D(G(x)),1)\)  where \(x \sim p_{data}\)  and \(L_{bce}\)  is the binary cross-entropy loss. The expression for \(L_R\)  and \(L_{KL}\)  are given in Equation  and Equation REF  respectively. Therefore, the objective that needs to be minimized by the VAE part is \(Loss = L_R + w_{KL}L_{KL} + L_{adv}^G\) .
\(L_{KL} & = & - \frac{1}{2N_z} \bigg (1 + \hat{\sigma } - \mu ^2 - exp(\hat{\sigma })\bigg )  \\L_R & = & L_s + L_p \quad \text{where, }  \\L_s & = & - \frac{1}{N_{max}} \sum _{i=1}^{N_s} log \big ( \mathcal {N}(\Delta x_i, \Delta y_i | \mu _{x,i}, \mu _{y,i}, \sigma _{x,i},\sigma _{y,i}) \big ) \nonumber \\L_p & = & - \frac{1}{N_{max}} \sum _{i=1}^{N_{max}} \sum _{k=1}^3 p_{k,i} log(q_{k,i}) \nonumber \) 
